Another prisoner puzzle

There are four prisoners, three situated one after another on a flight of stairs and the fourth is unseen behind an opaque wall. Prisoner 1 is at the top-left, Prisoner 2 is in the middle, and Prisoner 3 is at the bottom-right of the stairwell running from top-left to bottom-right. Prisoner 4 is behind the opaque wall. Prisoners 1 and 3 are wearing white hats while Prisoners 2 and 4 are wearing black ones.

Here’s an interesting prisoner problem. There are four prisoners, three on the stairs and one behind a wall. The prisoners on the stairs are placed so that the first prisoner can see the second’s and third’s hats, and the second can see only the third’s hat.

The jailer tells them there are four of them, that each is wearing a hat, and that there are two white and two black hats. The jailer says that they’ll all be free to go if even one of them guesses their hat colour right, but they’ll be sent straight to the gallows if any one of them guesses it wrong. The jailer also says that the prisoners are not allowed to communicate with each other in any way.

Well, the prisoners are all freed, so which prisoner was the one that guessed xir hat colour right and how did xe do it?

Please spend some time thinking about this question before reading the answer below. By the way the question is posed, you’ll know when you’ve got the right answer.

Answer: It helps to think about an alternate scenario. In this one, Prisoners 2 and 3 in the middle and bottom-right of the stairwell have the same hat colour. Since Prisoner 1 knows there are only two hats of the same colour, xir hat colour must be of the opposite colour. If this were the case, Prisoner 1 would have spoken. Prisoner 2 thinks this and wonders why Prisoner 1 hasn’t spoken yet. It must be because Prisoners 2 and 3 do not have the same hat colour, and Prisoner 1 does not wish to guess xir hat colour for fear of death. Once Prisoner 2 knows that xir hat colour is the opposite of Prisoner 3’s, xe has the answer. Prisoner 2 sees that Prisoner 3’s hat colour is white, so xirs must be black!

Therefore, the answer to our question is that Prisoner 2 guesses xir hat colour right.

Now, an interesting question to ask is if Prisoner 1 communicated to Prisoner 2 xir hat colour through xir silence… This raises questions about the nature of communication. This example shows that one can, under the right circumstances, communicate with just silence.

These kinds of puzzles are called induction puzzles. You can read more about them on Wikipedia.